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## G = C2×Q8○D8order 128 = 27

### Direct product of C2 and Q8○D8

direct product, p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C2×Q8○D8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C2×2- 1+4 — C2×Q8○D8
 Lower central C1 — C2 — C4 — C2×Q8○D8
 Upper central C1 — C22 — C2×C4○D4 — C2×Q8○D8
 Jennings C1 — C2 — C2 — C4 — C2×Q8○D8

Generators and relations for C2×Q8○D8
G = < a,b,c,d,e | a2=b4=e2=1, c2=d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d3 >

Subgroups: 996 in 704 conjugacy classes, 428 normal (12 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4○D4, C22×C8, C2×M4(2), C8○D4, C2×D8, C2×SD16, C2×Q16, C4○D8, C8.C22, C22×Q8, C22×Q8, C2×C4○D4, C2×C4○D4, C2×C4○D4, 2- 1+4, 2- 1+4, C2×C8○D4, C22×Q16, C2×C4○D8, C2×C8.C22, Q8○D8, C2×2- 1+4, C2×Q8○D8
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, C25, Q8○D8, D4×C23, C2×Q8○D8

Smallest permutation representation of C2×Q8○D8
On 64 points
Generators in S64
(1 60)(2 61)(3 62)(4 63)(5 64)(6 57)(7 58)(8 59)(9 36)(10 37)(11 38)(12 39)(13 40)(14 33)(15 34)(16 35)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 41)(24 42)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)(31 56)(32 49)
(1 28 5 32)(2 29 6 25)(3 30 7 26)(4 31 8 27)(9 17 13 21)(10 18 14 22)(11 19 15 23)(12 20 16 24)(33 48 37 44)(34 41 38 45)(35 42 39 46)(36 43 40 47)(49 60 53 64)(50 61 54 57)(51 62 55 58)(52 63 56 59)
(1 43 5 47)(2 44 6 48)(3 45 7 41)(4 46 8 42)(9 49 13 53)(10 50 14 54)(11 51 15 55)(12 52 16 56)(17 64 21 60)(18 57 22 61)(19 58 23 62)(20 59 24 63)(25 33 29 37)(26 34 30 38)(27 35 31 39)(28 36 32 40)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 8)(2 7)(3 6)(4 5)(9 16)(10 15)(11 14)(12 13)(17 24)(18 23)(19 22)(20 21)(25 30)(26 29)(27 28)(31 32)(33 38)(34 37)(35 36)(39 40)(41 44)(42 43)(45 48)(46 47)(49 56)(50 55)(51 54)(52 53)(57 62)(58 61)(59 60)(63 64)

G:=sub<Sym(64)| (1,60)(2,61)(3,62)(4,63)(5,64)(6,57)(7,58)(8,59)(9,36)(10,37)(11,38)(12,39)(13,40)(14,33)(15,34)(16,35)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,41)(24,42)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,49), (1,28,5,32)(2,29,6,25)(3,30,7,26)(4,31,8,27)(9,17,13,21)(10,18,14,22)(11,19,15,23)(12,20,16,24)(33,48,37,44)(34,41,38,45)(35,42,39,46)(36,43,40,47)(49,60,53,64)(50,61,54,57)(51,62,55,58)(52,63,56,59), (1,43,5,47)(2,44,6,48)(3,45,7,41)(4,46,8,42)(9,49,13,53)(10,50,14,54)(11,51,15,55)(12,52,16,56)(17,64,21,60)(18,57,22,61)(19,58,23,62)(20,59,24,63)(25,33,29,37)(26,34,30,38)(27,35,31,39)(28,36,32,40), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,8)(2,7)(3,6)(4,5)(9,16)(10,15)(11,14)(12,13)(17,24)(18,23)(19,22)(20,21)(25,30)(26,29)(27,28)(31,32)(33,38)(34,37)(35,36)(39,40)(41,44)(42,43)(45,48)(46,47)(49,56)(50,55)(51,54)(52,53)(57,62)(58,61)(59,60)(63,64)>;

G:=Group( (1,60)(2,61)(3,62)(4,63)(5,64)(6,57)(7,58)(8,59)(9,36)(10,37)(11,38)(12,39)(13,40)(14,33)(15,34)(16,35)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,41)(24,42)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,49), (1,28,5,32)(2,29,6,25)(3,30,7,26)(4,31,8,27)(9,17,13,21)(10,18,14,22)(11,19,15,23)(12,20,16,24)(33,48,37,44)(34,41,38,45)(35,42,39,46)(36,43,40,47)(49,60,53,64)(50,61,54,57)(51,62,55,58)(52,63,56,59), (1,43,5,47)(2,44,6,48)(3,45,7,41)(4,46,8,42)(9,49,13,53)(10,50,14,54)(11,51,15,55)(12,52,16,56)(17,64,21,60)(18,57,22,61)(19,58,23,62)(20,59,24,63)(25,33,29,37)(26,34,30,38)(27,35,31,39)(28,36,32,40), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,8)(2,7)(3,6)(4,5)(9,16)(10,15)(11,14)(12,13)(17,24)(18,23)(19,22)(20,21)(25,30)(26,29)(27,28)(31,32)(33,38)(34,37)(35,36)(39,40)(41,44)(42,43)(45,48)(46,47)(49,56)(50,55)(51,54)(52,53)(57,62)(58,61)(59,60)(63,64) );

G=PermutationGroup([[(1,60),(2,61),(3,62),(4,63),(5,64),(6,57),(7,58),(8,59),(9,36),(10,37),(11,38),(12,39),(13,40),(14,33),(15,34),(16,35),(17,43),(18,44),(19,45),(20,46),(21,47),(22,48),(23,41),(24,42),(25,50),(26,51),(27,52),(28,53),(29,54),(30,55),(31,56),(32,49)], [(1,28,5,32),(2,29,6,25),(3,30,7,26),(4,31,8,27),(9,17,13,21),(10,18,14,22),(11,19,15,23),(12,20,16,24),(33,48,37,44),(34,41,38,45),(35,42,39,46),(36,43,40,47),(49,60,53,64),(50,61,54,57),(51,62,55,58),(52,63,56,59)], [(1,43,5,47),(2,44,6,48),(3,45,7,41),(4,46,8,42),(9,49,13,53),(10,50,14,54),(11,51,15,55),(12,52,16,56),(17,64,21,60),(18,57,22,61),(19,58,23,62),(20,59,24,63),(25,33,29,37),(26,34,30,38),(27,35,31,39),(28,36,32,40)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,8),(2,7),(3,6),(4,5),(9,16),(10,15),(11,14),(12,13),(17,24),(18,23),(19,22),(20,21),(25,30),(26,29),(27,28),(31,32),(33,38),(34,37),(35,36),(39,40),(41,44),(42,43),(45,48),(46,47),(49,56),(50,55),(51,54),(52,53),(57,62),(58,61),(59,60),(63,64)]])

44 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 2K 2L 2M 4A ··· 4H 4I ··· 4T 8A 8B 8C 8D 8E ··· 8J order 1 2 2 2 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 8 8 8 8 8 ··· 8 size 1 1 1 1 2 ··· 2 4 4 4 4 2 ··· 2 4 ··· 4 2 2 2 2 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 D4 D4 D4 Q8○D8 kernel C2×Q8○D8 C2×C8○D4 C22×Q16 C2×C4○D8 C2×C8.C22 Q8○D8 C2×2- 1+4 C2×D4 C2×Q8 C4○D4 C2 # reps 1 1 3 3 6 16 2 3 1 4 4

Matrix representation of C2×Q8○D8 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 16 0 0 0 0 1 0 0 0 0 16 0 0 0 0 1 0 0 0
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 16 13 0 0 0 0 9 1 0 0 0 0 0 0 3 14 0 0 0 0 3 3 0 0 0 0 0 0 3 14 0 0 0 0 3 3
,
 16 13 0 0 0 0 0 1 0 0 0 0 0 0 3 3 0 0 0 0 3 14 0 0 0 0 0 0 14 14 0 0 0 0 14 3

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16,0,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[16,9,0,0,0,0,13,1,0,0,0,0,0,0,3,3,0,0,0,0,14,3,0,0,0,0,0,0,3,3,0,0,0,0,14,3],[16,0,0,0,0,0,13,1,0,0,0,0,0,0,3,3,0,0,0,0,3,14,0,0,0,0,0,0,14,14,0,0,0,0,14,3] >;

C2×Q8○D8 in GAP, Magma, Sage, TeX

C_2\times Q_8\circ D_8
% in TeX

G:=Group("C2xQ8oD8");
// GroupNames label

G:=SmallGroup(128,2315);
// by ID

G=gap.SmallGroup(128,2315);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,-2,448,477,456,521,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=e^2=1,c^2=d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d^3>;
// generators/relations

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